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Metamath Proof Explorer

Theorem clm0vs

Description: Zero times a vector is the zero vector. Equation 1a of Kreyszig p. 51. ( lmod0vs analog.) (Contributed by Mario Carneiro, 16-Oct-2015)

Ref Expression
Hypotheses clm0vs.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
clm0vs.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
clm0vs.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
clm0vs.z ⊢ 0 = ( 0g ‘ 𝑊 )
Assertion clm0vs ( ( 𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉 ) → ( 0 · 𝑋 ) = 0 )

Proof

Step Hyp Ref Expression
1 clm0vs.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 clm0vs.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3 clm0vs.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
4 clm0vs.z ⊢ 0 = ( 0g ‘ 𝑊 )
5 2 clm0 ⊢ ( 𝑊 ∈ ℂMod → 0 = ( 0g ‘ 𝐹 ) )
6 5 adantr ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉 ) → 0 = ( 0g ‘ 𝐹 ) )
7 6 oveq1d ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉 ) → ( 0 · 𝑋 ) = ( ( 0g ‘ 𝐹 ) · 𝑋 ) )
8 clmlmod ⊢ ( 𝑊 ∈ ℂMod → 𝑊 ∈ LMod )
9 eqid ⊢ ( 0g ‘ 𝐹 ) = ( 0g ‘ 𝐹 )
10 1 2 3 9 4 lmod0vs ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 0g ‘ 𝐹 ) · 𝑋 ) = 0 )
11 8 10 sylan ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 0g ‘ 𝐹 ) · 𝑋 ) = 0 )
12 7 11 eqtrd ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉 ) → ( 0 · 𝑋 ) = 0 )